Suppose $L \subset K$ are number fields and let's choose and fix an algebraic closure $\overline{L}$ of $L$ such that $L \subset K \subset \overline{L}$, hence $H:=\text{Gal}(\overline{L}/K)$ is a closed subgroup of $G:=\text{Gal}(\overline{L}/L)$ of finite index $[K, L]$. Suppose now we have an $\ell$-adic representation $V$ of $\text{Gal}(\overline{L}/K)$, then there will exist an induced representation $\text{Ind}_H^G\,V$ of $\text{Gal}(\overline{L}/L)$. If necessary we could assume $V$ has a geometric origin, e.g. it is the etale cohomology of a smooth variety defined over $K$.
For a finite prime $\mathfrak{P}$ of $K$, the local $L$-factor is defined as \begin{equation} L_{\mathfrak{P}}(V,s):=\text{det}(1-\text{Fr}_{\mathfrak{P}}(N(\mathfrak{P}))^{-s}\big|V^{I_{\mathfrak{P}}}) \end{equation} where $I_{\mathfrak{P}}$ is the inertia group at $\mathfrak{P}$. Similarly we could define the local $L$-factor for a prime $\mathfrak{p}$ of $L$. Question If $V$ is an Artin representation, we have \begin{equation} L_{\mathfrak{p}}(\text{Ind}_H^G\,V,s)=\prod_{\mathfrak{P}|\mathfrak{p}}L_{\mathfrak{P}}(V,s) \end{equation} I guess it is still true for $\ell$-adic representations, but I don't know how to prove. The method to prove this formula might still work, since the Galois groups $H$ and $G$ are locally compact, but I am not sure as I know basically nothing about representation theories of locally compact groups. It would be great if someone could give a complete proof.
